Gunn and Gott (1972)
suggested that the intracluster medium was primordial
intergalactic gas that had fallen into clusters. This intergalactic gas was
never associated with stars or galaxies, and thus could be expected to
have no heavy elements. They also noted that some of the intracluster
medium could come from ram pressure stripping of interstellar gas out of
galaxies. Assuming that clusters were immersed in uniform intergalactic
medium, they estimated the amount that would fall into clusters. By
comparing this to the amount of intracluster gas deduced from the early
X-ray observations of clusters, they could give an upper limit on the
density of the intergalactic gas. (This is an upper limit because some of
the intracluster gas could have come out of galaxies.) These limits are
usually given in terms of
c,
the density of matter necessary to close the universe

(5.120)

where h0 is the Hubble constant. If
IG
is the density of intergalactic gas, then Gunn and Gott found
IG
/ c 0.01. They used a
rather low value for the
gas mass in clusters, and more recent calculations (for example,
Cowie and Perrenod,
1978)
give
IG
/ c 0.2. Gunn and
Gott also noted that infall would
heat the intracluster gas to about the observed temperatures
(Section 5.3.2).

To determine the final configuration and evolution of the intracluster
gas in the infall model, hydrodynamic simulations of the infall have
been done by a number of authors
(Gull and Northover,
1975;
Lea, 1976;
Takahara et al.,
1976;
Cowie and Perrenod,
1978).
These calculations all assumed
that the cluster potential was fixed; the gas was assumed to fall into the
cluster after the cluster itself had collapsed. All of these
calculations were
one-dimensional simulations of spherical clusters, although a number of
different techniques were used to solve the hydrodynamic equations. With
the exception of Lea's calculations, these simulations have given similar
results.

As the gas first collapses into the core, its density increases and a shock
propagates outward from the cluster center and heats the gas. This shock
passes through the cluster in
109 yr,
essentially the sound crossing time
for the cluster (equation 5.54). After the passage of the shock, the hot
intracluster gas is nearly hydrostatic, and its further evolution is
quasistatic. As
the shock moves into the outer parts of the cluster, it weakens; less gas
is added to the cluster, and the cluster luminosity is nearly constant.
On the other hand, Lea found that the shock heating caused the gas
pressure to increase until the inflow was reversed and the intracluster gas
expanded. This cooled the gas adiabatically, lowered its pressure, and it
collapsed again. This process repeated itself, producing a large number of
pulsations with a period of about 5 × 109 yr. During
these pulsations,
the X-ray luminosity oscillated wildly between roughly 1041
and 1048
erg/s. The other calculations of the infall of intracluster gas have
failed to find these oscillations
(Gull and Northover,
1975;
Takahara et al.,
1976;
Cowie and Perrenod,
1978;
Perrenod, 1978b),
and they are probably
an artifact of Lea's numerical method. Such oscillations are in violent
disagreement with the observed X-ray luminosity function of clusters
(Schwartz, 1978).

Gull and Northover
(1975)
found that the shock strength was nearly constant
as it propagated outward; they argued that this occurred because the shock
speed was always about the free-fall speed in the cluster. They found
that the
resulting intracluster gas distribution was nearly adiabatic
(Section 5.5.2). On
the other hand, more detailed calculations by
Cowie and Perrenod
(1978) and
Perrenod (1978b)
found that the cluster gas distributions were not well
represented by any polytropic distribution, unless thermal conduction was so
effective that the gas was isothermal.

In the absence of significant cooling or thermal conduction,
Cowie and Perrenod
(1978)
showed that the infall models with a fixed cluster potential are
characterized by a single parameter, which gives the depth of the cluster
potential well, K
(r
/ H0rc)2 where
r and
rc are the cluster velocity
dispersion and core radius, respectively. Models with significant
cooling are also characterized by Btcoolh0, where the cooling time is
evaluated
at the cluster center. If thermal conduction is important, the cluster
evolution is also determined by the value of CTg / rcgcs3, where
is
the thermal conductivity, and Tg,
g,
and cs are the gas temperature,
density, and sound speed. When conduction saturates, the models are
independent of c. In general, the gas temperatures in these
models scale with
r2.

Cowie and Perrenod found that models without significant cooling or
conduction showed a very small decrease in the X-ray luminosity with time,
less than a 40% reduction from z = 1 to the present. This decrease in
luminosity resulted from the slow reexpansion of the intracluster gas
as the shock weakened. In models with significant cooling, the cluster
evolved to a nearly steady-state cooling flow
(Section 5.7). In models
with conduction, the X-ray luminosity increased slowly with time, by
about 40% from a redshift z = 1 to the present. This occurred because
conduction lowered the temperature in the cluster core
(Section 5.4.2). The
core then contracted so that the increasing density could maintain the
pressure in the core. Since the X-ray luminosity increases more rapidly
with density than does the pressure (equation 5.21), the luminosity went
up.

These models all assumed that the cluster potential was static; the cluster
was assumed to collapse before any gas fell into it. Of course, there is
no reason why intergalactic gas should wait until the cluster has formed
before gaseous infall can occur.
Perrenod (1978a,
b)
calculated the evolution of the intracluster
gas in infall models in which the cluster potential varied in time. The
cluster potential was taken from
White's (1976c)
N-body calculations of the collapse of a Coma-like cluster
(Section 2.9;
Figure 5). In White's
models, the cluster first collapses with violent relaxation, then contracts
slowly due to two-body interactions between galaxies. This contraction
causes the cluster potential well to become deeper, and as a result the
intracluster gas temperature and density increase with time. In contrast
to the
static potential models, Perrenod finds that the X-ray luminosity of his
model clusters increases by about an order of magnitude from z = 1 to
the present. The sizes of the gas distributions also shrink considerably.
If thermal conduction is important, the further contraction in the gas
distributions it produces makes them smaller than the observed sizes of
X-ray clusters.

One interesting aspect of Perrenod's models is that many of the infall
models
have a temperature inversion (dTg / dr > 0)
in the cluster core, even if there is no
significant cooling. This occurs because gas in the core has fallen
through a shallower gravitational potential than gas further out. If such a
temperature
inversion were observed, it might be confused with a cooling flow
(Section 5.7).

In White's N-body models, the cluster shows very strong subclustering at the
beginning of its collapse, and forms two roughly equal subclusters,
which merge as the cluster undergoes violent relaxation. Several double
X-ray clusters are known (Section 4.4.2;
Figure 18)
that appear to be in just this stage of evolution
(Forman et al.,
1981).
Obviously, such subclustering cannot be treated in
one-dimensional, spherical, hydrodynamic simulations.
Gingold and Perrenod
(1979)
have made simplified three-dimensional hydro simulations of the evolution
of clusters. When applied to the cluster potential from White's N-body
models, these verified the previous one-dimensional calculations of
Perrenod (1978b).
They found that there was no significant enhancement of the
X-ray emission from merging subclusters, beyond that predicted by single
cluster models. Similar calculations were made by
Ikeuchi and Hirayama
(1979).

One major concern about all the Perrenod varying-potential models is the use
of White's (1976c)
N-body calculations for the cluster potential. In this
particular set of models by White, the total virial mass of the cluster was
assumed to reside in the individual galaxies. This gave the galaxies
large masses, which increased their two-body interactions
(Section 2.9.1), and caused the
cluster core to contract rapidly. However, associating the missing mass
in clusters
with individual galaxies appears to produce more two-body relaxation in
clusters than is observed
(Sections 2.8 and
2.9.4);
in fact, this was one of
White's conclusions from his models. Thus it seems likely that Perrenod's
calculations may significantly overestimate the increase with time of the
X-ray luminosity and gas temperature and the decrease in the gas core
size.

Clusters of galaxies are the largest organized structures in the
universe, and
X-ray emission from them should be recognizable to large redshifts
(Chapter 6).
They might therefore be useful as probes of the cosmological structure
of the universe. Several cosmological tests have been proposed using
X-ray clusters
(Schwartz, 1976;
Silk and White, 1978);
although some of these tests
are relatively insensitive to X-ray cluster evolution, most are strongly
affected. These models suggest that it will be difficult to apply any tests
that require that X-ray clusters have remained unchanged since z = 1
(Perrenod, 1978b;
Falle and Meszaros,
1980).
On the other hand, in Perrenod's
models the luminosity and size of X-ray clusters depend strongly on
the density of material in the universe, since this determines the speed
with which clusters contract. In principle, this dependence of cluster
evolution on density might provide useful cosmological information; in
practice, the evolution models are too uncertain to be used reliably for
this purpose.

The models described so far have dealt with the evolution of single
clusters.
Perrenod (1980)
has attempted to predict the evolution of the luminosity
function of X-ray clusters
(Section 4.2). He assumed that galaxies
formed before
clusters, and that clusters were formed by the gravitational attraction
of galaxies. He argued that the merging of clusters tends to produce
larger clusters with deeper potential wells, and as a result the average
X-ray luminosity increases.
White (1982)
showed that this argument is
incorrect; the increase in the depth of cluster potential wells is more than
offset by the decrease in their characteristic densities. Perrenod found a
very rapid evolution of the luminosity function to higher luminosities; he
predicted that there should be few luminous X-ray clusters at redshifts
z
1/2. This evolution depends strongly on the average density of matter
in the universe, and Perrenod proposed using it as a cosmological test.
However, his basic model for clustering is apparently incorrect
(White, 1982).